Other Important Graphs and Equations
In addition to the graphs and equations of lines, the
Math IC will test your understanding of the graphs and equations
of parabolas and circles.
Questions on these topics will either ask you to match
the correct graph with the correct equation or give you an equation
and ask you to figure out certain characteristics of the graph.
Most of the questions about parabolas and circles are
straightforward. If you know the information in the sections below,
you’ll be able to breeze through them.
A parabola is a U-shaped curve that can open
either upward or downward.
A parabola is the graph of a quadratic function, which,
you may recall, is ax2 + bx + c.
The equation of a parabola can be expressed in two forms—the standard
form and the general form. Each can help you determine different
information about the nature of the parabola.
Standard Form of the Equation of a Parabola
The standard form of the equation of a parabola is perhaps
the most useful and will be the one most used on the Math IC test:
where a, h, and k are
constants. From this formula, you can learn a few pieces of information:
The vertex of the parabola is (h, k).
axis of symmetry of the parabola is the line x = h.
parabola opens upward if a > 0, and
downward if a < 0.
For example, if you were given the parabola equation y =
–3(x – 5)2 + 8, you
first need to pick out the values of the constants a, h,
and k. Then you can derive information
about the parabola. For this example, a =
–3, h = 5, and k =
8. So the vertex is (5, 8), the axis of symmetry is the line x =
5, and since –3 < 0, the parabola opens downward.
General Form of the Equation of a Parabola
The general form of the equation of a parabola is:
where a, b, and c are
constants. If a question presents you with a parabola equation in
this form, you can find the following information about the parabola:
The vertex of the parabola is (–b
/2a, c –
axis of symmetry of the parabola is the line x =
parabola opens upward if a > 0, and downward if a <
- The y-intercept
is the point (0, c).
A circle is the collection of points equidistant
from a given point, called the center of the circle. For the Math
IC test, there is only one equation you have to know for a circle.
This equation is called the standard form:
where (h, k) is the
center of the circle, and r is the radius. When
the circle is centered at the origin, so that h = k =
0, then the equation simplifies to:
That’s it. That’s all you need to know about a circle
in coordinate geometry. Once you know and understand this equation,
you should be able to sketch a circle in its proper place on the
coordinate system if you are given its equation. You will also be
asked to figure out the equation of a circle if you are given a
picture of its graph.
To test your knowledge, try to answer the following practice
is the equation of the circle pictured below?
The center is given in the image: (–2, –1). All you need
to finish the formula is the radius. We determine this by finding
the distance from the center and the point, (2, –4), pictured on
The radius of the circle is 5, so the equation of the
circle can be written as (x + 2)2 +
(y + 1)2 = 25.